RSMA-Aided Satellite-Aerial-Vehicle Integrated Networks: Secrecy Performance Evaluation with Non-ideal Hardware

I. INTRODUCTION

WITH the improvement of the wireless communication systems, higher energy efficiency and higher security have been recognized as the bottomed requirements for the communication systems [1]–[5]. Under these considerations, satellite communication (SatCom) system appears due to its characters, i.e., wide coverage and reliable communication quality [6], [7]. However, owing to several reasons, the satellite sometimes can not transmit the signal to the destination user directly, which requires the assistance of the aerial nodes [8]. The aerial nodes can be moving or fixed by the form of the aerial type, such as high aerial platform (HAP) or unmanned aerial vehicle (UAV) [9]. Relied on this foundations, the satellite-aerial-vehicle integrated network (SAVIN) comes to our sight owing to its own characters, which combines both the advantages of the SatCom and the vehicle communications, thus it has attracted many interests in recent years [10]–[12]. The SAVIN has been confirmed as the hopeful framework for the next generation networks, especially for the sixthgeneration networks [13], [14]. The SAVIN is also recognized the important comment of the real networks, namely, the digital video broadcasting (DVB) systems and the Integrated Space-Ground Information Network Engineering of China [15], [16].

A. Related Works

The research of SAVIN has been a famous issue in recent twenty years owing to the development of the electronic devices. In the first two decade of the 21st century, some researchers have studied this issue. The authors in [17] researched the outage probability (OP) of the SAVINs, besides, the authors got the exact expressions for the OP. In [18], the authors focused on the beamforming problem for the SAVINs. In [19], the symbol error ratio (SER) was investigated for the SAVINs. However, the former papers have derived the closedform expressions by using the some finite series functions, which are hard to follow and to be calculated. At this time, the authors of [20] have simplified the analysis for the SAVINs and obtained the widely used channel model for the satellite transmission link, which is the very popular one being utilized in so many lately works. In [21], the authors utilized this channel model and proposed a max-max scheduling strategy for the SAVINs. In [22], the authors researched the secondary networks performance for the SAVINs under cognitive technology by applying this channel model. The authors of [23] studied the OP for the amplify-and-forward (AF) SAVINs, where multiple users and this popular channel model were both considered.

For some reasons, the transmission networks are not always ideal, they often suffer from non-ideal limitations, such as high non-linearity, phase noise and the other factors, which form the non-ideal hardware (NIH) of the communication nodes [24]–[26]. The authors of [27] researched the influence of NIH on the SatCom systems. The authors of [28] studied the influence of NIH on the SAVINs, especially, the partial relay selection strategy was utilized, moreover, the exact expressions for the OP were further got. In [29], a relay selection strategy was used by setting the proper threshold to gain a balance which means the system can get the acceptable performance by applying proper system complexity, particularly, the detailed investigations for the OP were obtained. In [30], the influence of NIH was investigated for the cognitive SAVIN, especially, the analysis of OP was further obtained. In [31], multiple UAVs were considered for the SAVINs by considering the influence of NIH. In addition, the OP was further researched. In [32], the influences of NIH and simultaneous wireless information and power transfer (SWIPT) on the SAVINs were researched. The authors of [33] studied the influence of NIH nodes on the SAVINs with adaptive relay protocol and cognitive technology, particularly, the system performance was further investigated. In [34], the influence of NIH on the cognitive SAVINs was investigated by considering a direct transmission link. In [35], the impact of NIH on the OP for the SAVINs was further investigated. In [36], the authors investigated the SER for the considered SAVIN by considering the impact of non-orthogonal multiple access (NOMA) and NIH.

As mentioned before, higher security is one basic requirement of the next generation systems, moreover, owing to the inherent factors of the satellite, the SatCom often has a wide coverage, which will lead to the existence of the secrecy issues, especially for the SAVINs. In order to investigate the secrecy issue in theoretical level, physical layer security (PLS) comes to our sight, which can analyze the security problem in theoretical level, besides, it has been regarded to be the hot topic these years [37]–[42]. In [42], the authors concluded all the security issues for the SatCom and proposed the future directions for the secrecy issue of the SatComs. The authors in [43] proposed a user scheduling strategy by applying the opportunistic scheme for the legitimate users and obtained the exact investigations for the secrecy outage probability (SOP), particularly, the probability density function (PDF) for the shadowed-Rician (SR) channel under opportunistic scheduling scheme was first derived. In [44], the influence of NIH was evaluated on the SatCom for the secrecy problems, particularly, the exhaustive investigation for the SOP was further obtained. In [14], the influence of NIH was researched for the SAVINs with several relays, particularly, the two-way technology was applied in it. In [45], the authors researched the SOP for the reconfigurable intelligent surface (RIS)-based the SAVINs, in addition, to enhance the secrecy performance, maximal ratio combining (MRC) scheme was utilized to estimate the secrecy system performance.

Just presented before, another requirement for the next generation is higher energy efficiency, based on this requirement, NOMA is proposed to exhaustive the energy efficiency [46], [47]. The goal of NOMA scheme is to guarantee multiple users simultaneously and allocate the frequency resources by assuming different signals owning different power levels [48]. Through [49], the influence of NIH was studied on the SAVINs with NOMA technique, especially, the exact expression for the SOP was derived along with the asymptotic analysis. In [8], the influences of NIH and NOMA scheme were evaluated for the cognitive SAVINs. In [50], the authors maximized the sum secrecy rate for the NOMA-based cognitive SAVINs. In [51], PLS of a NOMA-aided downlink SAVIN was researched.

B. Motivations

NOMA scheme has its own characters, however, NOMA scheme needs perfect successive interference cancellation (SIC) technology in each legitimate user [52]. To reduce the system complexity, rate-splitting multiple access (RSMA) comes to our sight [53]. When considering the RSMA strategy, every user’s information is separated into two forms, one is common information and the other one is private information. The common information comes from common signals and the private message comes from the private signal. The common signal will be transmitted to all users with SIC technique. The private signals will be forwarded to each corresponding user by taking the other users’ private signals as noise [54]. On this foundation, SIC technique is only utilized once, which will lead to a reduced complexity when compared to the NOMA strategy [55], [56]. On this foundation, RSMA has been regarded as a hopeful multiple access way [57]. RSMA has also been used into the SatCom network. Through [58], the influence of RSMA was researched on the SatCom systems by utilizing the cognitive technique. In [59], the authors studied the OP for the RSMA-based SAVINS in the presence of relay selection strategy. The authors of [60] studied the energy efficiency problem for the RSMA-based SatCom networks, particularly, multiple groups and multiple beams were considered. In [61], the authors investigated the optimization method to get max-min rate fairness for the SAVINs in the presence of RSMA scheme. Some optimization methods have been used to gain a better system performance for the SAVINs [62]–[65]. As the authors realize that only few papers have analyzed the secrecy problem for the RSMA-based SAVINs. In [66], the authors used the optimization method to enlarge the secure transmission rate for the RSMA-based SAVINs along with the cognitive technology. In [67], the authors investigated security and energy problem in the cognitive SAVINs with RSMA. However, from the public reports that there are few open papers or literatures both investigating the influences of NIH and RSMA on the SOP for the SAVINs, which motivates the contributions of this work.

C. Main Works

Inspired from the former descriptions, by both considering the RSMA and NIH into our view, we take a representative SAVIN for the consideration. Particularly, the main contributions are shown as

By both considering the RSMA and NIH into consideration, we give a general secrecy SAVIN model, which comprises a satellite, multiple UAVs, several destinations and several eavesdroppers with non-colluding eavesdropping scheme.

Relied on the secrecy system network, the analysis for the SOP is further derived along with the exact expressions. Through these expressions, the SOP can be evaluated better. Besides, the influence of NIH on the SOP can be even judged.

The asymptotic derivations for the SOP are further investigated along with the simple and efficient expressions in high signal-to-noise ratio (SNR) scenario. Moreover, the secrecy coding gain (SCG) and secrecy diversity order (SDO) are also gotten.

The analysis for the secrecy energy efficiency (SEE) is further investigated for the regarded secrecy networks, especially the comparisons with the NOMA scheme.

Several representative Monte Carlo (MC) simulations are plotted to confirm the efficiency of the theoretical derivations along with some interesting findings and results.

The left parts of this work are shown as: In Section II, the introduction of the secrecy SAVIN model is provided. The exact expressions and asymptotic analysis for SOP are presented along with the SCG and SDO in Section III. Though Section IV, the investigations of SEE are provided to study the energy efficiency. In Section V, several representative MC results are shown to confirm the validity of the former analysis. At last, during Section VI, a summary for the whole work is provided.

Notations: [TeX:] $$E[\cdot]$$ denotes the expected function dedenotes the expected function defined in [68], [TeX:] $$(\cdot)_{\mathrm{x}}$$ represents the Pochhammer symbol [68, page xliii]. [TeX:] $$|\cdot|$$ represents the absolute value [68], [TeX:] $$B(., .)$$ represents the Beta function [68, Eq. 8.384.1].

II. SYSTEM MODEL AND PROBLEM FORMULATION

As plotted in Fig. 1, through the work, we provide a general secrecy SAVIN into account, which comprises one satellite, multiple UAVs [TeX:] $$R_p, p \in\{1, \cdots, P\},$$ multiple legitimate users [TeX:] $$D_j, j \in\{1, \cdots, M\},$$ and multiple potential eavesdroppers [TeX:] $$E_q, q \in\{1, \cdots, L\}$$¹. Moreover, each transmission node is assumed to own one antenna². Moreover, [TeX:] $$R_p$$ works at hallduplex with decode-and-forward (DF) protocol³. The entire transmission needs two time slots. For the first time slot, the S forwards its signal to R, during the second time slot, R retransmits the transmitted signal to the legitimate destinations via DF mode.

¹ Besides, during this work, the direct transmission link between the S and [TeX:] $$D_j$$ is not available for the reason that shadowing and blocking.

² Although in this paper, only one antenna is assumed, our system model and obtained results are also suitable for the case with multiple antenna with a suitable beamforming scheme.

³ In this paper, owing to the long distance of the satellite to the UAV, thus DF protocol is applied at the UAV to enhance the signal transmission.

Through the first time slot, RSMA is utilized at S, where the information of each user can be divided into two comments, namely, one is common comment and the other is private comment. The common factors for all users are included in the common information, which are encoded in the common information signal [TeX:] $$s_c(t).$$ On the other hand, the private message for each user will be contained in the private signal [TeX:] $$s_{p,j}(t).$$ Besides, we have the following assumption as [TeX:] $$\overline{R}_{c, j}=\overline{R}_c$$ and [TeX:] $$\overline{R}_{p, j}=\overline{R}_p$$, where [TeX:] $$\bar{R}_{c, j}$$ represents the transmission rate for the common signal for [TeX:] $$D_j.$$ [TeX:] $$\bar{R}_{p, j}$$ represents the target rate of the private signal for [TeX:] $$D_j.$$ It is noted that, the target rate of [TeX:] $$D_j$$ with its form as [TeX:] $$\overline{R}_j=\overline{R}_{c, j}+\overline{R}_{p,j}.$$

The transmitted signal s (t), can be defined as

where [TeX:] $$a_j \text{ and } a_c$$ represent the allocation elements of power for the private signal and the common signal of [TeX:] $$D_j$$, respectively. In addition, [TeX:] $$E\left[|s(t)|^2\right]=E\left[\left|s_c(t)\right|^2\right]=E\left[\left|s_{p, j}(t)\right|^2\right]=1$$ is assumed. Moreover,

Then, the signal gotten at the pth R is presented as

where [TeX:] $$P_S$$ depicts the transmission power of S, [TeX:] $$h_{S R_p}$$ denotes the channel element from S to the pth R with shadowing as SR fading, s (t) represents the transmission signal which has been shown in (1), [TeX:] $$\eta_{S R_p}(t)$$ depicts the distortion noise which comes from the NIH with shadowing as [TeX:] $$\eta_{S R_p}(t) \sim \mathcal{C N}\left(0, \kappa_{S R_p}^2\right), \kappa_{S R_p}$$ is the level of NIH at the pth R. [TeX:] $$n_{S R_p}(t)$$ depicts the additive white Gaussian noise (AWGN) at the pth R with its expression as [TeX:] $$n_{S R_p}(t) \sim \mathcal{C N}\left(0, \delta_{S R_p}^2\right).$$

At the pth R, the common signal [TeX:] $$s_c(t)$$ is decrypted by recognizing the private signals of the legitimate users as interference or noise. By focusing on this consideration, the signal-to-interference-and-noise plus distortion ratio (SINDR) for the pth R by decrypting the common signal [TeX:] $$s_c(t)$$ has the following expression as

where [TeX:] $$\lambda_{S R_p}=\left|h_{S R_p}\right|^2 P_S / \delta_{S R_p}^2.$$

When we decode [TeX:] $$s_c(t)$$ successfully, the decoded [TeX:] $$s_c(t)$$ which SIC is applied will be deleted among the obtained signal. The private signal for [TeX:] $$s_{p . j}(t)$$ is decrypted by regarding the other legitimate users’ private signal as interferences. Relied on this consideration, the SINDR for the pth R by decoding the private signal [TeX:] $$s_{p . j}(t)$$ is obtained as

For the second time slot, the chosen R re-forwards the obtained signal to the legitimate users with the DF mode. Thus, our obtained signal at [TeX:] $$D_j$$ has the following expression as

where [TeX:] $$P_R$$ depicts the transmitted power of the selected R, [TeX:] $$h_{R_p D_j}$$ represents the channel factor between the pth R and the jth D with shadowing as the Rayleigh fading. [TeX:] $$\eta_{R_p D_j}(t)$$ denotes the distortion noise following [TeX:] $$\eta_{R_p D_j}(t) \sim \mathcal{C N}\left(0, \kappa_{R_p D_j}^2\right), \kappa_{R_p D_j}$$ is the NIH level at the jth D, [TeX:] $$n_{D_j}(t)$$ depicts the AWGN at the jth D shadowed as [TeX:] $$n_{D_j}(t) \sim \mathcal{C N}\left(0, \delta_{R D_j}^2\right).$$

When applying the same way of [TeX:] $$S \rightarrow R$$ link and (6), the SINDR of the legitimate user [TeX:] $$D_j$$ for decoding [TeX:] $$s_c(t)$$ has the expression as

where [TeX:] $$\lambda_{R_p D_j}=\left|h_{R_p D_j}\right|^2 P_R / \delta_{R_p D_j}^2.$$

When SIC is utilized in the considered system, [TeX:] $$s_c(t)$$ will be removed from the signal received, the SINDR for the legitimate user [TeX:] $$D_j$$ by decoding [TeX:] $$s_{p . j}(t)$$ is written as

As the wider coverage of the UAV and satellite beam, thus the secrecy issue often exists in the SAVIN. Owing to each eavesdropper can steal the legitimate information, the derived signal of the qth eve is derived as

where [TeX:] $$h_{R_p E_q}$$ represents the channel coefficient between the pth R and the qth E, [TeX:] $$\eta_{R_p E_q}(t)$$ denotes the distortion noise which follows [TeX:] $$\eta_{R_p E_q}(t) \sim \mathcal{C N}\left(0, \kappa_{R_p E_q}^2\right), \kappa_{R_p E_q}$$ is the NIH level at the qth E, [TeX:] $$n_{E_q}(t)$$ denotes the AWGN at the qth E shadowed as [TeX:] $$n_{E_q}(t) \sim \mathcal{C N}\left(0, \delta_{R_p E_q}^2\right).$$

From (9), the signal-to-noise plus distortion ratio (SNDR) for the qth E is given by

where [TeX:] $$\lambda_{R_p E_q}=\left|h_{R_p E_q}\right|^2 P_R / \delta_{R_p E_q}^2 .$$

In the former discussion, non-colluding eavesdropping strategy is applied for the secrecy system, then the eventual SNDR of the eavesdropping link has the following expression as

Secrecy capacity has been investigated in so many former works, in this paper, we used the definition in [14], which uses the discrepancy between the eavesdroppers’ link and the legitimate users’ link. Then, with the assistance of (7), (8), and (11), the secrecy capacity of the whole network is obtained as

where [TeX:] $$C_{B_j}=\log _2\left(1+\gamma_{D_{j c}}+\gamma_{D_{j p}}\right), C_E=\log _2\left(1+\gamma_E\right) \text {, }$$ and [TeX:] $$[x]^{+} \triangleq \max [x, 0].$$

III. SECRECY PERFORMANCE EVALUATION

Through the following parts, the secrecy performance will be researched. At the very beginning, the PDF and cumulative distribution function (CDF) of the considered vehicle transmission channel and satellite transmission channel model are given in the following.

A. Channel Model

1) The vehicle channel model: As mentioned in the former section, the channel model between the UAV and the legitimate users or the eavesdroppers undergoes the Rayleigh fading. Then, the PDF and CDF for [TeX:] $$\lambda_{G G}, G G \in\left\{R_p D_j, R_p E_q\right\}$$ are, respectively, written as

and

where [TeX:] $$\overline{\lambda}_{G G}$$ represents the average channel gain.

By using [49] and (10), (11), (13), (14), the CDF and PDF for [TeX:] $$\gamma_E$$ are, respectively, written as

and

2) The satellite channel model: Through the considered secrecy system, the geosynchronous earth orbit (GEO) satellite⁴ is studied. Moreover, multiple transmission beams are considered for the GEO in the presence of time division multiple access (TDMA) scheme [27]. Relied on the consideration, only one UAV is served for transmitting the signal for each time slot. The channel factor [TeX:] $$h_{S R_p}$$ from the UAV to the kth downlink on-board beam of satellite, which is regarded as

where [TeX:] $$f_{S R_p}$$ denotes the SR factor of the satellite transmission link. [TeX:] $$C_{S R_p}$$ depicts the effects of the free loss (FSL) and antenna pattern with the expression as

where c represents the velocity of the light, f depicts the frequency of the transmitted wave, l represents the length between the UAV and the on-board beam center of the kth beam. [TeX:] $$l_0 \approx 36000 \mathrm{~km} . G_{S R_p}$$ depicts the antenna gain between the satellite and the UAV. [TeX:] $$G_{R e}$$ represents the on-board kth beam gain of the satellite.

⁴ We take the GEO satellite for the analysis, however the derived analysis is also utilized to case with medium Earth orbit (MEO) and low Earth orbit (LEO) satellites.

With the help of [14], [TeX:] $$G_{R e}$$ can be derived as

where [TeX:] $$\bar{G}_{\max }$$ is known as the largest beam gain at the boresight and depicts the angle of the off-boresight. When analyzing the [TeX:] $$G_{R e}$$, from [29], the antenna gain between the satellite and the UAV of the kth beam has the expression as

where [TeX:] $$G_m$$ represents the maximal beam gain, [TeX:] $$\xi_k=2.07123 \sin \vartheta_k / \sin \bar{\vartheta}_k, \vartheta_k$$ is the angle between UAV position and the on-board beam center from the kth beam, [TeX:] $$\bar{\vartheta}_k$$ denotes the 3 dB angle for the kth beam. [TeX:] $$K_1 \text{ and } K_3$$ depicts the 1st kind bessel function with order 1 and 3, respectively. [TeX:] $$\vartheta_k$$ is considered to be zero, which leads to the result that [TeX:] $$G_{S R_p} \approx G_m.$$ Relied on the above considerations, we can get [TeX:] $$h_{S R_p}=C_{S R_p}^{\max } f_{S R_p},$$ where

By investigating [TeX:] $$f_{S R_p},$$ the authors in [69] proposed a general utilized SR model, which is suitable for the mobile SatCom [14]. From [29], the channel element [TeX:] $$f_{S R_p}$$ has the following form as [TeX:] $$f_{S R_p}=\bar{f}_{S R_p}+\widetilde{f}_{S R_p},$$ where the scattering factor [TeX:] $$\widetilde{f}_{S R_p}$$ is considered to undergo the independent and identically distribution (i.i.d.) Rayleigh fading, while [TeX:] $$\bar{f}_{S R_p}$$ represents the element of line of sight (LoS) factor with undergoing i.i.d. Nakagami-m fading.

On this foundation, the PDF of [TeX:] $$\lambda_{S R_p}=\bar{\lambda}_{S R_p}\left|C_{S R_p}^{\max } f_{S R_p}\right|^2$$ can be written as

where [TeX:] $$\overline{\lambda}_{S R_p}$$ is the average SNR between the UAV and the satellite. [TeX:] $$\Delta_{S R_p}=\frac{\beta_{S R_p}-\sigma_{S R_p}}{\overline{\lambda}_{S R_p}},$$ [TeX:] $$\alpha_{S R_p}=\left(\frac{2 b_{S R_p} m_{S R_p}}{2 b_{S R_p} m_{S R_p}+\Omega_{S R_p}}\right)^{m_{S R_p}} / 2 b_{S R_p}$$ [TeX:] $$\delta_{S R_p}=\frac{\Omega_{S R_p}}{\left(2 b_{S R_p} m_{S R_p}+\Omega_{S R_p}\right) 2 b_{S R_p}},$$ [TeX:] $$\beta_{S R_p}=1 / 2 b_{S R_p},$$ where [TeX:] $$m_{S R_p} \geq 0, \Omega_{S R_p} \text { and } 2 b_{S R_p}$$ are the channel elements, which has the meanning as the fading severity parameter, the LoS element’s average power and the multipath’s component’s power. Under this popular assumption, [TeX:] $$m_{S R_p}$$ is regarded to be an integer through this work [22]. Eventually, [TeX:] $$(\cdot)_{k_1}$$ depicts the Pochhammer symbol [68]. When [TeX:] $$m_{S R_p} \rightarrow \infty,$$ the shadowed-Rician channel becomes to the Rician fading.

After applying (22) and utilizing [21], the CDF of [TeX:] $$\lambda_{S R_p}$$ is re-given by

B. Secrecy Outage Probability

Through this work, SOP is defined as the probability that the SINDR of either transmission link is smaller than a target threshold, which has the definition as

where [TeX:] $$P_1\left(\gamma_0\right)$$ represents the OP for the [TeX:] $$S \rightarrow R$$ transmission link, [TeX:] $$P_2\left(\gamma_0\right)$$ denotes the SOP for the [TeX:] $$R \rightarrow D$$ transmission link.

Theorem 1. The SOP of the considered system has the following expression as

where [TeX:] $$P_{\text {out } 2}\left(\gamma_0\right)=1$$ and

with [TeX:] $$A=1-a_c^2+\kappa_{S R_p}^2, B=\sum_{\substack{i=1 \\ i \neq j}}^M a_j^2+\kappa_{S R_p}^2,$$ [TeX:] $$A_1=1-a_c^2+\kappa_{R_p D_j}^2, \text { and } B_1=\sum_{\substack{i=1 \\ i \neq j}}^M a_j^2+\kappa_{R_p D_j}^2.$$ [TeX:] $$P_{\text {out } 1}\left(\gamma_0\right)$$ is presented as (26), which is found at the top of next page. In (26), [TeX:] $$H_1=\min \left\{\frac{1}{\kappa_{R_p E_q}^2}, \frac{a_c^2 B_1+a_j^2 A_1}{A_1 B_1}\right\}$$ U represents the number of terms, [TeX:] $$y_i=t_i+1$$ is the ith zero of Legendre polynomials, [TeX:] $$\omega_i$$ denotes the Gaussian weight with its derivation in Table (25.4) of [70].

In (26), g (y) has the expression as

where [TeX:] $$F_{\gamma_{B_1}}(x) \text { and } F_{\gamma_{B_2}}(x)$$ are written as (28) and (29), shown at the top of next page.

Proof. See Appendix A.

C. Asymptotic SOP

In order to gain the influences of system parameters and channel parameters on the SOP at high SNRs. In this subsection, the asymptotic investigations for SOP in high SNR regime are derived.

In this subsection, we assume that [TeX:] $$\overline{\lambda}_{R_p D_j}=\overline{\lambda}_{S R_p}=\overline{\gamma},$$ when [TeX:] $$\bar{\gamma} \rightarrow \infty \text {, }$$ (14) and (23) are re-written, respectively, as

and

where o (x) represents the higher order of x.

Theorem 2. The asymptotic SOP for the system in high SNR regime is given as (32) shown in the middle of next page.

In (32), h (y) is defined as

where [TeX:] $$F_{\gamma_{B_3}}(x) \text { and } F_{\gamma_{B_4}}(x)$$ are written as (34) and (35), which are shown at the 8th page, respectively. In (33), [TeX:] $$U_1$$ represents the number of terms, [TeX:] $$y_\tau=t_\tau+1$$ depicts the th zero of Legendre polynomials, [TeX:] $$\nu_\tau$$ denotes the Gaussian weight with derivations in Table (25.4) of [70].

Proof. By replacing (14) and (23) with (30) and (31), then using the similar way of Theorem 1, (32) will be derived.

The proof is over.

To derive SDO and SCG, (32) has the following form,

where

and the SCG can be derive as

IV. SECRECY ENERGY EFFICIENCY

For the practical reasons, batteries or the solar panels are the major power forms of the satellite and UAVs. Thus, the SEE is often utilized to verify the efficiency of the secrecy systems. From [8] and [49], the SEE has the following definition as

where [TeX:] $$R_{S Y}$$ depicts the secrecy capacity of the transmission systems, v represents the amplifier efficiency of the power. [TeX:] $$P_{int}$$ depicts the cost of the fixed power, such as the circuit power and the other left power. From the former discussions, [TeX:] $$R_{S Y}$$ can be written as [TeX:] $$R_{S Y}=C_{S Y}=\sum_{j=1}^M\left[1-P_{1 p}\left(\gamma_0\right)\right]\left(C_{B_j}-C_E\right),$$ so, (39) can be re-given by

V. NUMERICAL RESULTS

In the following, the theoretical analysis is attested by the MC results. Generally, the GEO satellite is taken as an example⁵, and [TeX:] $$\delta_{S R_p}^2=\delta_{R_p D_j}^2=\delta_{R_p E_q}^2=1, \bar{\lambda}_{S R_p}=\bar{\lambda}_{R D}=\bar{\gamma}.$$ The system parameters are shown in Table I [28] along with channel parameters shown in Table II [14], respectively. Besides, we set [TeX:] $$\kappa_{S R_p}=\kappa_{R_p D_j}=\kappa_{R_p E_q}=k .$$ Figs. 2–5 use three power allocation Cases. In Case I, [TeX:] $$a c^2=0.79 ;$$ In Case II, [TeX:] $$a c^2=0.82 ;$$ In case III, [TeX:] $$a c^2=0.85$$.

⁵ The derivations in this works are still suitable for the scene with the other orbit satellite.

Fig. 2 plots the SOP versus different [TeX:] $$\bar{\gamma}$$ with M=3, L=3, [TeX:] $$\bar{\gamma}_E=-5 \mathrm{~dB} \text {, }$$ P=3 and [TeX:] $${\gamma}_0=-4 \mathrm{~dB}$$ in several shadowing scenarios. Fig. 2, it can be seen the MC simulations are tight across the theoretical results through the whole SNRs. Besides, when in high SNR scenarios, the asymptotic derivations are nearly the same with MC results, which indicate the rightness of the theoretical and asymptotic derivations. Next, we can observe that, when the channel is under light fading, the SOP will be lower, which indicates that the SOP is mainly decided by the satellite to UAV link. Moreover, it can be found that when the transmission nodes suffer from heavy impairments, the SOP will be larger, which has been factor of the NIH system [8], [14].

Fig. 3 introduces the SOP versus different [TeX:] $$\bar{\gamma}$$ and different [TeX:] $$\gamma_0$$ with M=3, L=3, [TeX:] $$\bar{\gamma}_E=-5 \mathrm{~dB} \text {, }$$ and P=3 for FHS scenario. From Fig. 3, it can be derived that some same results with Fig. 2, thus, we do not introduce them again to save the space of the paper. Furthermore, when the threshold of the secrecy system grows larger, the SOP will become larger, too, which is the inherent factor of the secrecy system [49], [71].

Fig. 4 plots the SOP versus different [TeX:] $$\bar{\gamma}$$ and different P with M=3, L=3, [TeX:] $$\bar{\gamma}_E=-5 \mathrm{~dB} \text {, }$$ and [TeX:] $${\gamma}_0=-4 \mathrm{~dB}$$ for FHS scenario. Obtained from Fig. 4, when P becomes smaller, the SOP will be higher as a results that less UAVs are utilized for the transmission. However, when we refer to this figure again, we can find that all the curves are parallel, which show that SDO of the considered system is same. From the former analysis, this MC simulations and the theoretical results are the same.

Fig. 5 depicts SOP versus different [TeX:] $$\bar{\gamma}$$ and different [TeX:] $$\bar{\gamma}_E$$ with M=3, L=10, P=3, and [TeX:] $${\gamma}_0=-4 \mathrm{~dB}$$ for FHS scenario. In this figure, we can know that when the power of [TeX:] $$\bar{\gamma}_E$$ become smaller, the SOP will be lower, too,. This is because that the ability of the eavesdropper is powerful, it can overhear much information.

Fig. 6 illustrates the SOP versus different [TeX:] $$\bar{\gamma}$$ and different power allocation factors with M=3, L=3, P=3, [TeX:] $$\bar{\gamma}_E=-5 \mathrm{~dB} \text {, }$$ and [TeX:] $${\gamma}_0=-4 \mathrm{~dB}$$ for FHS scenario. Derived in Fig. 5, we can obtain that when the system utilizes the allocation scheme with Case I, the SOP will be lower when compared with Case II. This indicates that we need to allocate much power to the common signal rather than that of private signals.

Fig. 7 examines the SOP versus [TeX:] $$\bar{\gamma}=20 \mathrm{~dB}$$ and different [TeX:] $$\gamma_0$$ with M=3, L=3, and P=3 for FHS and AS scenario. Interestingly, it can be derived that, if the secrecy system suffers from the NIH, the SOP will be always 1 with the threshold increasing to a special value. This value is just the function of the NIH level regardless of the other reasons. A smaller NIH level will lead to a larger value. Besides, when compared with two shadowing scenarios, we can find that the certain constant is the same, which is not decided by the channel fading and the own characters of NIH system.

Fig. 8 plots the SOP versus different [TeX:] $$\bar{\gamma}$$ and different selection schemes with M=3, L=3, [TeX:] $$\bar{\gamma}_E=-5 \mathrm{~dB} \text {, and } \gamma_0=-4 \mathrm{~dB}$$ for FHS scenario. It can be found that the secrecy performance of the proposed scheme (partial selection scheme) is superior to that of random selection scheme for the reason that when using partial selection scheme during the simulation results, the best UAV is selected, however, when applying the random selection scheme, one random UAV is chosen. Obviously, the performance of random selection is worse than that of partial selection scheme. Moreover, it is interesting that when the system undergoes light NIH, the SOP will be lower, which is the inherent character of the NIH systems [8], [49].

From Fig. 2 to Fig. 8, it can be obtained that the channel fading, the NIH level, the threshold, the [TeX:] $$\bar{\gamma}_E$$, the number of UAVs, the number of eavesdroppers and selection scheme have serious impacts on the SOP of the considered secrecy system.

Figs. 9–11 illustrate the SEE for the system withM=2, L=2, [TeX:] $$\bar{\gamma}_E=-5 \mathrm{~dB} \text {, and } \gamma_0=-4 \mathrm{~dB}$$ From [72], [TeX:] $$v=2 \text { and } P_{\text {int }}=5 \mathrm{~dBW}$$ are considered for the whole system. Derived by Fig. 9, we can find that much power is allocated to the common message, the SEE will be larger. Obtained from Fig. 10, when the network is suffering from heavy channel fading, the SEE will be lower. However, the satellite shadowing has little influence on the SEE according to the MC simulation results. It maybe come from that the UAV uses the DF protocol to enhance the secrecy performance. From Fig. 11, we can find that the SEE of RSMA is superior to that of NOMA scheme, which is the advantage of the RSMA scheme. It is interesting that all the curves in these three figures, all the SEEs have an extreme point. When the [TeX:] $$\bar{\gamma}$$ is larger than this point, the SEE will be lower as the [TeX:] $$\bar{\gamma}$$ becomes larger. Interestingly, we find that when [TeX:] $$\bar{\gamma}$$ is smaller than this point, the SEE will be larger as [TeX:] $$\bar{\gamma}$$ becoming larger.

VI. CONCLUSIONS

This paper researched the secrecy performance for the RSMA-based SAVINs with the existence of the NIH and multiple eavesdroppers. When considering the secrecy networks, the entire expressions for the SOP were further derived. In addition, the MC simulations were obtained to confirm the the correctness of the theoretical analysis. The results indicated that less P, the heavy channel fading, much eavesdropping power, larger NIH level and larger secrecy threshold would lead to a worse system performance. Besides, the simulation results for the SEE were also provided. Interestingly, we found that there existed a turning value for the SEE. Finally, much power was allocated to the common signal would bring a more secrecy system.

APPENDIX A

PROOF OF LEMMA 1

From (24), it can be found that the important thing to get the derivations for [TeX:] $$P_1\left(\gamma_0\right) \text { and } P_2\left(\gamma_0\right).$$ In what follows, the exact expressions for them will be gotten.

Firstly, the detailed process for obtaining [TeX:] $$P_1\left(\gamma_0\right)$$ will be provided. By utilizing the partial selection scheme, the OP for the [TeX:] $$S \rightarrow R$$ link, i.e., [TeX:] $$P_1\left(\gamma_0\right)$$ can be written as

where

where [TeX:] $$P_{1 p c}\left(\gamma_0\right)$$ denotes the OP for the pth R by decoding the common message and [TeX:] $$P_{1 p p}\left(\gamma_0\right)$$ represents the OP for the pth R by decoding the private message.

By utilizing (23) and (4), [TeX:] $$P_{1 p c}\left(\gamma_0\right)$$ is derived as

With the same way of deriving [TeX:] $$P_{1 p c}\left(\gamma_0\right)$$, the closed expression for [TeX:] $$P_{1 p p}\left(\gamma_0\right)$$ is given by

Next, after inserting (43) and (44) into (42), [TeX:] $$P_{1 p}\left(\gamma_0\right)$$ will be derived. After inserting the expression of [TeX:] $$P_{1 p}\left(\gamma_0\right)$$ into (41), the closed-form expression for [TeX:] $$P_{1 p}\left(\gamma_0\right)$$ will be got.

The second thing is to get the closed-form expression for [TeX:] $$P_2\left(\gamma_0\right)$$.

From (12), [TeX:] $$P_2\left(\gamma_0\right)$$ can be written as

where [TeX:] $$C_0=\log _2\left(1+\gamma_0\right).$$

In (45), [TeX:] $$J_1$$ is the important one which requires to be obtained. [TeX:] $$J_1$$ can be written as

In (46), it is important to get the CDF of [TeX:] $$F_{\gamma_{D_{j c}}+\gamma_{D_{j p}}}(x),$$ from (7) and (8), [TeX:] $$F_{\gamma_{D_{j c}}+\gamma_{D_{j p}}}(x)$$ can be re-given by

where [TeX:] $$x_2=\frac{-E-\sqrt{E^2+4 D x}}{2 D}, x_1=\frac{-E+\sqrt{E^2+4 D x}}{2 D},$$ [TeX:] $$D=a_c^2 B_1+a_j^2 A_1-x A_1 B_1 \text { and } E=a_c^2+a_j^2-x\left(A_1+B_1\right) \text {. }$$

Then, by using (14), (28) and (29) could be obtained.

Then, by submitting (47) and (16) into (46), (46) can be re-derived as

However, trying the author’s best knowledge, (48) can not be derived in regular method. In addition, with the assistance of [70] and utilizing the Gaussian-Chebyshev quadrature, [TeX:] $$J_1$$ has the following expression as

In addition, by inserting (49) into (45), the expression for [TeX:] $$P_2\left(\gamma_0\right)$$ can be obtained.

Finally, by inserting [TeX:] $$P_1\left(\gamma_0\right) \text { and } P_2\left(\gamma_0\right)$$ into (24), after some calculating steps, the SOP can be obtained.

Proof is finished.